Tool CaT
stdout:
MAYBE
Problem:
plus(plus(X,Y),Z) -> plus(X,plus(Y,Z))
times(X,s(Y)) -> plus(X,times(Y,X))
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(plus(X, Y), Z) -> c_0(plus^#(X, plus(Y, Z)))
, 2: times^#(X, s(Y)) -> c_1(plus^#(X, times(Y, X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {2}, Uargs(c_0) = {}, Uargs(times^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 2] x1 + [1 0] x2 + [0]
[0 2] [0 1] [2]
times(x1, x2) = [2 3] x1 + [2 1] x2 + [0]
[3 2] [3 0] [0]
s(x1) = [1 2] x1 + [3]
[0 0] [2]
plus^#(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [3 3] x1 + [2 3] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {times^#(X, s(Y)) -> c_1(plus^#(X, times(Y, X)))}
Weak Rules:
{ times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
Proof Output:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {}, Uargs(times^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[4 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
times^#(x1, x2) = [7 7] x1 + [2 0] x2 + [7]
[7 7] [0 0] [7]
c_1(x1) = [2 0] x1 + [3]
[0 0] [3]
* Path {2}->{1}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {2}, Uargs(c_0) = {1}, Uargs(times^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 1] x1 + [1 0] x2 + [1]
[0 2] [0 1] [3]
times(x1, x2) = [1 3] x1 + [1 2] x2 + [0]
[2 2] [2 0] [0]
s(x1) = [1 1] x1 + [2]
[0 1] [3]
plus^#(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {plus^#(plus(X, Y), Z) -> c_0(plus^#(X, plus(Y, Z)))}
Weak Rules:
{ times^#(X, s(Y)) -> c_1(plus^#(X, times(Y, X)))
, times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
Proof Output:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(times^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 1] x1 + [0 0] x2 + [2]
[2 0] [0 0] [2]
times(x1, x2) = [4 4] x1 + [2 4] x2 + [0]
[4 0] [2 0] [4]
s(x1) = [1 4] x1 + [0]
[0 0] [2]
plus^#(x1, x2) = [4 2] x1 + [0 0] x2 + [0]
[6 0] [4 0] [0]
c_0(x1) = [2 0] x1 + [5]
[0 0] [6]
times^#(x1, x2) = [7 7] x1 + [2 0] x2 + [6]
[7 7] [0 0] [7]
c_1(x1) = [1 0] x1 + [3]
[0 0] [3]Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
YES(?,O(n^2))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X))}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^2))
Input Problem: runtime-complexity with respect to
Rules:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X))}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: plus^#(plus(X, Y), Z) -> c_0(plus^#(X, plus(Y, Z)))
, 2: times^#(X, s(Y)) -> c_1(plus^#(X, times(Y, X)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{1} [ YES(?,O(n^2)) ]
Sub-problems:
-------------
* Path {2}: YES(?,O(n^1))
-----------------------
The usable rules for this path are:
{ times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {2}, Uargs(c_0) = {}, Uargs(times^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 2] x1 + [1 0] x2 + [0]
[0 2] [0 1] [2]
times(x1, x2) = [2 3] x1 + [2 1] x2 + [0]
[3 2] [3 0] [0]
s(x1) = [1 2] x1 + [3]
[0 0] [2]
plus^#(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
times^#(x1, x2) = [3 3] x1 + [2 3] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {times^#(X, s(Y)) -> c_1(plus^#(X, times(Y, X)))}
Weak Rules:
{ times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
Proof Output:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {}, Uargs(times^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[4 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
plus^#(x1, x2) = [0 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
times^#(x1, x2) = [7 7] x1 + [2 0] x2 + [7]
[7 7] [0 0] [7]
c_1(x1) = [2 0] x1 + [3]
[0 0] [3]
* Path {2}->{1}: YES(?,O(n^2))
----------------------------
The usable rules for this path are:
{ times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(plus) = {2}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {2}, Uargs(c_0) = {1}, Uargs(times^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [0 1] x1 + [1 0] x2 + [1]
[0 2] [0 1] [3]
times(x1, x2) = [1 3] x1 + [1 2] x2 + [0]
[2 2] [2 0] [0]
s(x1) = [1 1] x1 + [2]
[0 1] [3]
plus^#(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {plus^#(plus(X, Y), Z) -> c_0(plus^#(X, plus(Y, Z)))}
Weak Rules:
{ times^#(X, s(Y)) -> c_1(plus^#(X, times(Y, X)))
, times(X, s(Y)) -> plus(X, times(Y, X))
, plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))}
Proof Output:
The following argument positions are usable:
Uargs(plus) = {}, Uargs(times) = {}, Uargs(s) = {},
Uargs(plus^#) = {}, Uargs(c_0) = {1}, Uargs(times^#) = {},
Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
plus(x1, x2) = [2 1] x1 + [0 0] x2 + [2]
[2 0] [0 0] [2]
times(x1, x2) = [4 4] x1 + [2 4] x2 + [0]
[4 0] [2 0] [4]
s(x1) = [1 4] x1 + [0]
[0 0] [2]
plus^#(x1, x2) = [4 2] x1 + [0 0] x2 + [0]
[6 0] [4 0] [0]
c_0(x1) = [2 0] x1 + [5]
[0 0] [6]
times^#(x1, x2) = [7 7] x1 + [2 0] x2 + [6]
[7 7] [0 0] [7]
c_1(x1) = [1 0] x1 + [3]
[0 0] [3]